Question

what is the argument of (1-sin theta)+icos theta​

Answer 1

Argument = π/4   + θ/2

Step-by-step explanation:

Z = (1 - Sinθ) + i Cosθ

argument θ

Tanθ    =    Cosθ/(1 - Sinθ)

Using Cos2x = Cos²x - Sin²x

& Cos²x + Sin²x = 1

& Sin2x = 2SinxCosx

=> Tanθ  = (Cos²(θ/2) - Sin²(θ/2))/( Cos²(θ/2) +Sin²(θ/2)  - 2Sin(θ/2)Cos(θ/2)

Using a² - b² = (a + b)(a - b)

=> Tanθ  =  ((Cos(θ/2) + Sin(θ/2) (Cos(θ/2) - Sin(θ/2))/(Cos(θ/2) - Sin(θ/2)²

=>  Tanθ  =  (Cos(θ/2) + Sin(θ/2))/(Cos(θ/2) - Sin(θ/2)

=>  Tanθ  =  ( 1 + Tan(θ/2))/( 1 - Tan(θ/2))

Using Tan(π/4) = 1

=>  Tanθ  =  ( Tan(π/4) + Tan(θ/2))/( 1 - Tan(π/4)Tan(θ/2))

Using Tan(A + B) = (TanA + TanB)/(1 - TanATanB)

=>  Tanθ  = Tan(π/4   + θ/2)

=> θ  = π/4   + θ/2

Argument = π/4   + θ/2

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